Minimal Fibre Structure of the Non-Injective Projection from Born–Infeld Indiscernability: Derivation of the Parity Involution

The identification of conjugate Weil blocks ,\, q-c\ as the fibres of the non-injective projection Π:χ→O was stated as a structurally motivated hypothesis in O16 and used in O17 to derive the pair-level observable and the exponent doubling δₚₐᵢᵣ = 2\,δc ≈ 7.44. The present paper provides the foundational derivation of the underlying involution. We proceed in two stages. At the abstract level, we show that the Born–Infeld action S[χ] is even in χ, and we define a notion of BI-indiscernability for configurations that produce identical effective responses under all BI-admissible perturbations. Evenness of S immediately implies that χ and -χ are BI-indiscernible, so every projective fibre contains the orbit \χ, -χ\. We then prove a conditional minimality result: in the absence of any symmetry of S[χ] beyond its parity, the minimal fibre is exactly the involution χ↦-χ. At the concrete level, the O17 result ρq-c = ρc together with the norm-invariance of Gram–Schmidt orthogonalisation identifies the involution c↔ q-c as the Weil-representation instance of the abstract parity. This closes the logical gap in O16–O17 by deriving the fibre structure from the Born–Infeld parity rather than postulating it.